Blowup of a scheme along an ideal sheaf

Assume the Axiom of Choice as inherited from the relative Proj construction (The Axiom of Choice). Let $X$ be a scheme and let $\mathcal{I}\subseteq {\mathcal{O}}_X$ be a quasi-coherent ideal sheaf of finite type (Quasi-coherent ideal sheaves), with zero scheme $Z=V(\mathcal{I})$, the closed subscheme of $X$ cut out by $\mathcal{I}$. The blowup of $X$ along $\mathcal{I}$ (or along $Z$) is the $X$-scheme

${\mathrm{Bl}}_{\mathcal{I}}X:={\mathrm{Proj}}_X\mathcal{R}(\mathcal{I}),$

the relative Proj of Relative Proj of a graded quasi-coherent algebra applied to the Rees algebra sheaf $\mathcal{R}(\mathcal{I})={⨁}_{n\ge 0}{\mathcal{I}}^n$ of Rees algebra sheaf of a finite type ideal, equipped with its structural morphism

$\pi \text{\hspace{0.17em}⁣}:{\mathrm{Bl}}_{\mathcal{I}}X⟶X$

and its relative twists $\mathcal{O}(n)$, $n\in \mathbb{Z}$, both as in Relative Proj of a graded quasi-coherent algebra. The notation records the ideal sheaf $\mathcal{I}$, not merely the closed subscheme $Z$; the finite type hypothesis is part of the definition because the later structural results for blowups (invertibility of the pullback of $\mathcal{I}$, the exceptional divisor, and base change) are proved under it.

In the affine case $X=\mathrm{Spec}A$ with $\mathcal{I}=\tilde{I}$ for an ideal $I\subseteq A$, the Rees algebra sheaf is $\mathcal{R}(\mathcal{I})\cong \widetilde{{⨁}_{n\ge 0}{I}^n}$ (Rees algebra sheaf of a finite type ideal), and the absolute case of the relative Proj construction identifies ${\mathrm{Bl}}_{\mathcal{I}}X$ with the absolute Proj of the Rees algebra $R(I)={⨁}_{n\ge 0}{I}^n{t}^n$ (Relative Proj of a graded quasi-coherent algebra); the structural morphism is then the Proj structural morphism to $\mathrm{Spec}A$.

The exceptional subscheme of the blowup is denoted $E$ and is introduced separately; no property of $E$ is assumed here.

• The Axiom of Choice is inherited from the affine-local Proj construction used by Relative Proj of a graded quasi-coherent algebra; the blowup selects no further data beyond that interface.
• This item only sets up the construction. Projectivity of $\pi$, the universal property of the blowup, the invertibility of the pullback of $\mathcal{I}$, and flat base change are supplied by later items of this page and are not asserted here.